Microfractured media with a scale and MumfordShah energies.pdf

arxiv 0704 3791v1 math ap 28 apr 2007 microfractured media with a scale and mumford shah energies marius buliga 03 2007 abstract we want to understand he concentration of damage in microfractured elastic media due to the diff erent scallings of the volume and area or area and length in two dimensions the traditional method of homogenization using periodic arrays of cells seems to fail when applied to the mumford shah functional and to periodically fractured domains in the present paper we are departing from traditional homogenization the main result implies the use of mumford shah energies and leads to an explanation of the observed concentration of damage in microfractured elastic bodies 1introduction a new direction of research in brittle fracture mechanics begins with the article of mumford shah 15 regarding the problem of image segmentation this problem which consists in fi nding the set of edges of a picture and constructing a smoothed version of that picture it turns to be intimately related to the problem of brittle crack evolution in the before mentioned article mumford and shah propose the following variational approach to the problem of image segmentation let g r2 0 1 be the original picture given as a distribution of grey levels 1 is white and 0 is black let u r be the smoothed picture and k be the set of edges k represents the set where u has jumps i e u c1 k r the pair formed by the smoothed picture u and the set of edges k minimizes then the functional i u k z u 2dx z u g 2dx h1 k the parameter controls the smoothness of the new picture u controls the l2 distance between the smoothed picture and the original one and controls the total length of the edges given by this variational method the authors remark that for 0 the functional i might be useful for an energetic treatment of fracture mechanics simion stoilow institute of mathematics of the romanian academy po box 1 764 014700 bucharest romania e mail marius buliga imar ro 1 an energetic approach to fracture mechanics is naturally suited to explain brittle crack appearance under imposed boundary displacements the idea is presented in the followings the state of a brittle body is described by a pair displacement crack u k is such a pair if k is a crack seen as a surface which appears in the body and u is a displacement of the broken body under the imposed boundary displacement i e u is continuous in the exterior of the surface k and u equals the imposed displacement u0 on the exterior boundary of the body let us suppose that the total energy of the body is a mumford shah functional of the form e u k z w u dx f u0 k the fi rst term of the functional e represents the elastic energy of the body with the displacement u the second term represents the energy consumed to produce the crack k in the body with the boundary displacement u0as parameter then the crack that appears is supposed to be the second term of the pair u k which minimizes the total energy e models for brittle damage based on functionals of the mumford shah type have have been proposed by francfort marigo 11 buliga 6 among others such models have been studied intensively from the mathematical point of view espacially by the italian school of geometric measure theory to name a few de giorgi ambrosio dal maso buttazzo the fi rst homogenization result concerning the mumford shah functional seems to be braides defranceschi vitali 5 in this paper it is done the homogenization of a mumford shah functional of the form z f x u d z su g x u u u dhn 1 the paper focardi gelli 14 and the references therein are part of another line of research which might be relevant for this paper homogenization of perforated domains in the present paper we are departing from traditional homogenization the line of research concerning perforated domains is close to our problem but for various reasons the results from perforated domains don t apply here we want to understand he concentration of damage in microfractured elastic media due to the diff erent scallings of the volume and area or area and length in two dimen sions the traditional method of homogenization using periodic arrays of cells seems to fail when applied to the mumford shah functional and to periodically fractured domains the main result theorem 4 2 implies the use of mumford shah energies and leads to an explanation of the observed concentration of damage in microfractured elastic bodies instead of performing a homogenization of the total energy of the microfractured body and then study the minimizers of the homogenized energy we proceed along a diff erent path we study sequences of problems on fractured elastic bodies indexed by 2 a scale parameter each such problem has at least approximative solutions we fi nd estimates of the area of the damaged region in terms of the scale 2notations let be a bounded open subset of r2 with locally lipschitz boundary we denote by y 0 1 2the unit closed square in r2 for a given 0 let z r2be the lattice of points in r2with coordinates of the form m n for all m n z we denote by z z the set of all z z such that z y to any z z we associate the cell dz z y the set z is fi nite for any 0 we denote the cardinal of this set by n and we notice that as goes to 0 we have lim 0 n 2 a 1 where a denotes the area of thus for small the number of cells n is approximately equal to a 2 3the model we take to be the confi guration set of a microfractured linear elastic body we explain further what we mean by this the elastic properties of the body are described by an elastic potential w m2 2 sym r r we suppose that the function w is quadratic and strictly positive defi nite for a given displacement u r2 the elastic energy of the body is given by z w e u dx where e u is the deformation of the displacement u that is the symmetric part of the gradient of u for any x e u x 1 2 u x u t x for a fi xed 0 we suppose that the body contains a distribution of micro fractures at the scale seen as a union of lipschitz curves f z z z fz 3 where for each z z the lipschitz curve fzlies inside the unit cell y fz 0 1 2 we explain further what we mean by an imposed boundary displacement u0 and what we mean by u u0on the boundary of we consider for simplicity that u0 rnis a continuous and therefore bounded function then for any u sbd u u0if the approximate limit of u equals u0 in any point of where the fi rst exists i e for all x if there exists v x such that lim o r b x u y v x dy b x 0 then v x u0 x defi nition 3 1 the class of admissible displacements with respect to the distribution of cracks f and with respect to the imposed displacement u0 is defi ned as the collection of all u sbd such that a u u0on b f su this class of admissible displacements is denoted by adm f u0 this defi nition deserves an explanation an admissible displacement u is a function which has to be equal to the imposed displacement on the boundary of condition a any such function u is a special function with bounded deformation that is a reasonably smooth function on the set suand the function u is allowed to have jumps along the set su for the technical details see the appendix we have to think about su as being a collection of curves with fi nite length physically the set surepresents the collection of all cracks in the body under the displacement u the condition b tells us that the collection of all cracks associated to an admissible displacement u contains f at least defi nition 3 2 with the notations from defi nition 3 1 the total energy of an admis sible displacement u adm f u0 is given by e u z w e u dx gh1 su f the energy of an admissible displacement is of mumford shah type it contains two terms the fi rst term measures the elastic energy of the body under the displacement u notice that in the expression of the elastic energy we have integrated over the whole 4 domain this is simply because the collection of cracks associated to u that is the set su has lebesque measure 0 therefore we have z w e u dx z su w e u dx in physical terms the right hand side expression would make more sense than the left hand side but from the mathematical point of view they are the same this is not meaning that the elastic energy neglects the fractures indeed further we shall infi mize the energy e over the whole set of admissible displacements according to condition b of defi nition 3 1 this set is defi ned with respect to the collection of cracks f therefore the infi mum of the energy e depends on the set of cracks f the second term of the mumford shah energy measures the surface energy caused by the apparition of new cracks the collection of new cracks is the set su f the constant g has the dimension of energy per unit area and it is physically related to the griffi th constant in 4 has been proven that functionals like e are l1inferior semi continuous and coercive hence on closed subspaces v of sbd the functional e has a mini mizer such a closed subspace of sbd is the space of all admissible displacements adm f u0 therefore we have theorem 3 3 on the space adm f u0 we consider the topology given by the con vergence uh u if uhl2 u hn 1 suh su 0 then there exists a minimizer of the functional e over the set adm f u0 in the following section we shall use approximate minimizers defi nition 3 4 for a given 0 a function u adm f u0 is a approximate minimizer if e u inf e v v adm f u0 for fi xed 0 we model an approximate displacement of a microfractured body as a sequence of displacements u with converging to 0 such that for each 0 the displacement u adm f u0 is a approximate minimizer of the mumford shah energy e over the set adm f u0 notice that in the model at this stage there is no relation between the crack sets f f for two diff erent scales 4an estimate related to damage concentration for fi xed 0 given f and imposed boundary displacement u0 let u adm f u0 be a approximate minimizer of the mumford shah energy e 5 in this section we want to estimate the number of cells z y z z where the initial cracks z fzpropagated let l 0 be a given length defi nition 4 1 for any cell dz z y z z and any approximate mini mizer u we defi ne the emergent crack in the cell dzby su z z y su z fz a cell dzis called active if the length of the emergent crack is greater than l that is h1 su z l we denote by m l the number of active cells in this notation we don t mention the dependence of m l on the approximate minimizer u theorem 4 2 suppose that for fi xed 0 the crack sets f are chosen so that there exists an approximate displacement of a microfractured body u with converging to 0 with the property that the sequence inf e v v adm f u0 is bounded then the number of active cells m l is of order 1 and the area of the damaged region of the body damaged dzactive dz is of order proof let m 0 such that for all 0 we have inf e v v adm f u0 m according to defi nition 3 4 for any 0 we have e u z w e u dx gh1 su f inf e v v adm f u0 m from defi nition 4 1 we get the following estimate h1 su f x z z h1 su z m l l we have therefore g m l l gh1 su f e u m 6 all in all we have obtained the estimate m l 1 m gl the area of the damaged region of the body is area damaged x dzactive area dz 2m l m gl the proof is done 5conclusions the theorem implies that the area of the damaged region is much smaller than the total area of the body as goes to zero in this model the use of mumford shah energies leads to an explanation of the observed concentration of damage in microfractured elastic bodies notice that we need more precise estimates in order to prove that the damaged region at the scale converges as goes to zero to a curve with fi nite length all we know at this moment is that the area of the damaged region goes to zero as the scale parameter in experiments it has been observed that the damaged region is approximately straight it is possible that mumford shah energies might explain this since geometries of the active crack set that is su f close to a straight line would be preferred by the energy e see 7 for examples that in some situations the leading term of a mumford shah energy is the one accounting for the length of the crack and not the elastic energy part finally in theorem 4 2 we obtained an estimate of the number of cells where cracks of length at least l appear it would be interested to study the interplay between and l in this estimate 6appendix functions with bounded variation or defor mation this section is dedicated to a brief voyage trough the spaces sbv and sbd the space sbv rn of special functions with bounded variation was introduced by de giorgi and ambrosio in the study of a class of free discontinuity problems 9 1 2 for any function u l1 rn let us denote by du the distributional derivative of u seen as a vector measure the variation of du is a scalar measure defi ned like this for any borel measurable subset b of the variation of du over b is du b sup x i 1 du ai i 1ai b ai aj i 6 j 7 a function u has bounded variation if the total variation of du is fi nite we send the reader to the book of evans gariepy 13 for basic properties of such functions the space sbv rn is defi ned as follows sbv rn u l1 rn du dsu s u 0 the lebesgue set of u is the set of points where u has approximate limit the com plementary set is a lnnegligible set denoted by su if u is a special function with bounded variation then su is also i e countably rectifi able from the calderon zygmund 8 decomposition theorem we obtain the following expression of du the distributional derivative of u sbv rn seen as a measure du u x dx u n dhn 1 k we shall use further the notation if the measure is absolutely continuous with respect to the measure let us defi ne the following sobolev space associated to the crack set k see 3 w 1 2 k u sbv rn z u 2dx z k u 2dhn 1 dsu hn 1 k it has been proved in 10 the following equality w1 2 k rn l rn w 1 2 k rn l rn 6 0 1 a similar description can be made for the space of special functions with bounded deformation sbd can be found in 4 for any function u l1 rn we denote by eu the symmetric part of the distributional derivative of u seen as a vector measure we denote also by ju the subset of where u has diff erent approximate limits with respect to a point dependent direction the diff erence between suand juis subtle let us quote only the fact that for a function u sbv rn the diff erence of these sets is hn 1 negligible the defi nition of sbd is the following sbd rn u l1 rn eu 0 then u is countably rectifi able ju uand hn 1 u ju 0 theorem 6 2 let u l1 rm then calderon zygmund if u bv rm then u is approximately diff erentiable ln a e in the approximate diff erential map x 7 u x is integrable du splits into three mutually singular measures on du u dx u hn 1 su cu where u is the jump of u in respect with the normal direction on su cu is the cantor part of du defi ned by cu a dsu a su where dsu is the singular part of du in respect to ln belletini coscia dal maso let m n and u bd then u has symmetric approximate diff erential u ln a e in and eu splits into three mutually singular measures on eu u dx u hn 1 ju ecu moreover u is approximately diff erentiable ln a e in theorem 6 3 the following are true w1 1 rm bv rm the inclusion is continuous in respect with the banach space topologies if u sbv rm then u w 1 1 su rm moreover if u w1 1 k rm l rm where k is a closed countably rectifi able set with hn 1 k then u sbv rm and hn 1 k su 0 let le1 be the banach space of l1 rn functions with l1symmetric dif ferential if u sbd then u le1 ju let k be a closed count ably rectifi able set with hn 1 k if u le1 k l rn then u sbd and hn 1 k ju 0 9 references 1 l ambrosio variational problems in sbv and image segmentation acta appl mathematic 17 1989